Algebra Functions and Graphing

Functions

• Domain and Range:
  • Domain: set of all input values (x) for which the function is defined
  • Range: set of all output values (y) that the function can produce
• Function Operations:
  • Sum/Difference: (f ± g)(x) = f(x) ± g(x)
  • Product: (fg)(x) = f(x) * g(x)
  • Composition: (f ∘ g)(x) = f(g(x))
• Function Properties:
  • Even/Odd: f(-x) = f(x) or f(-x) = -f(x)
  • Increasing/Decreasing: f(x) ≥ f(y) or f(x) ≤ f(y) for x > y

Graphing

• Graph Types:
  • Linear: straight line, y = mx + b
  • Quadratic: parabola, y = ax^2 + bx + c
  • Exponential: y = a * b^x
  • Trigonometric: y = a * sin(x) or y = a * cos(x)
• Graph Transformations:
  • Vertical Shift: y = f(x) ± k
  • Horizontal Shift: y = f(x ± h)
  • Reflection: y = -f(x) or y = f(-x)
• Graphing Techniques:
  • Plotting points
  • Using symmetry
  • Identifying asymptotes

Trigonometry

• Angles and Triangles:
  • Degree measure: 0° ≤ θ ≤ 360°
  • Radian measure: 0 ≤ θ ≤ 2π
  • Pythagorean Identity: sin^2(θ) + cos^2(θ) = 1
• Trigonometric Functions:
  • Sine: sin(θ) = opposite side / hypotenuse
  • Cosine: cos(θ) = adjacent side / hypotenuse
  • Tangent: tan(θ) = opposite side / adjacent side
• Trigonometric Identities:
  • Sum and Difference Formulas
  • Double and Half Angle Formulas

Analytic Geometry

• Coordinate Systems:
  • Cartesian Coordinates (x, y)
  • Polar Coordinates (r, θ)
• Equations of Lines:
  • Slope-Intercept Form: y = mx + b
  • Point-Slope Form: y - y1 = m(x - x1)
  • Standard Form: Ax + By = C
• Circles and Conic Sections:
  • Circle: (x - h)^2 + (y - k)^2 = r^2
  • Ellipse: (x - h)^2/a^2 + (y - k)^2/b^2 = 1
  • Parabola: y = a(x - h)^2 + k

Sequences and Series

• Sequences:
  • Arithmetic Sequence: an = a1 + (n - 1)d
  • Geometric Sequence: an = a1 * r^(n - 1)
• Series:
  • Arithmetic Series: Σan = (a1 + an)/2 * n
  • Geometric Series: Σan = a1 * (1 - r^n) / (1 - r)
• Convergence Tests:
  • nth Term Test
  • Ratio Test
  • Root Test

Functions

• Domain of a function refers to the set of all input values (x) for which the function is defined.
• Range of a function refers to the set of all output values (y) that the function can produce.
• Function operations include sum/difference, product, and composition, with formulas (f ± g)(x) = f(x) ± g(x), (fg)(x) = f(x) * g(x), and (f ∘ g)(x) = f(g(x)) respectively.
• Functions can have even or odd properties, where f(-x) = f(x) or f(-x) = -f(x) respectively.
• Functions can also be increasing or decreasing, where f(x) ≥ f(y) or f(x) ≤ f(y) for x > y.

Graphing

• Linear graphs are straight lines represented by the equation y = mx + b.
• Quadratic graphs are parabolas represented by the equation y = ax^2 + bx + c.
• Exponential graphs are represented by the equation y = a * b^x.
• Trigonometric graphs are represented by the equations y = a * sin(x) or y = a * cos(x).
• Graph transformations can be vertical shifts (y = f(x) ± k), horizontal shifts (y = f(x ± h)), or reflections (y = -f(x) or y = f(-x)).
• Graphing techniques include plotting points, using symmetry, and identifying asymptotes.

Trigonometry

• Angles can be measured in degrees (0° ≤ θ ≤ 360°) or radians (0 ≤ θ ≤ 2π).
• The Pythagorean Identity is sin^2(θ) + cos^2(θ) = 1.
• Sine (sin), cosine (cos), and tangent (tan) are trigonometric functions, where sin(θ) = opposite side / hypotenuse, cos(θ) = adjacent side / hypotenuse, and tan(θ) = opposite side / adjacent side.
• Trigonometric identities include sum and difference formulas, and double and half angle formulas.

Analytic Geometry

• Coordinate systems can be Cartesian (x, y) or polar (r, θ).
• Equations of lines can be in slope-intercept form (y = mx + b), point-slope form (y - y1 = m(x - x1)), or standard form (Ax + By = C).
• Circles can be represented by the equation (x - h)^2 + (y - k)^2 = r^2.
• Ellipses and parabolas are types of conic sections, represented by the equations (x - h)^2/a^2 + (y - k)^2/b^2 = 1 and y = a(x - h)^2 + k respectively.

Sequences and Series

• Arithmetic sequences have the form an = a1 + (n - 1)d, while geometric sequences have the form an = a1 * r^(n - 1).
• Arithmetic series have the formula Σan = (a1 + an)/2 * n, while geometric series have the formula Σan = a1 * (1 - r^n) / (1 - r).
• Convergence tests for series include the nth term test, ratio test, and root test.